Page 353 - Contributed Paper Session (CPS) - Volume 2

CPS1885 Karuna G. R. et al.
∞
, > 0. (10)
Γ() = ∫ −1 −
0

The objective function in (5) could be expressed as a function of boundary
points and ℎ−1 where = − ℎ−1 ; ≥ 0, ℎ = 1, 2, . . . , denotes
ℎ
ℎ
ℎ
ℎ
the width of the h th stratum within the range of the distribution, = − .
Using (5), the problem given in (4) could be expressed as an MPP as given
below:
+
2
{ ( + 1) [ (, ℎ−1 ) − (, ℎ−1 ℎ )]

+
Minimize ∑ × [ ( + 2, ℎ−1 ) − ( + 2, ℎ−1 ℎ )]

ℎ=1
+ 2
− [ ( + 1, ℎ−1 ) − ( + 1, ℎ−1 ℎ ) ] × }
2 2
{ ℎ }

subject to ∑ = ,
ℎ
ℎ=1

and ≥ 0; ℎ = 1,2, … , . (11)
ℎ
where = − = − , and are parameters of the Gamma

0
distribution, and (·) is the Upper Regularized Incomplete Gamma function.
To solve the MPP (11) using the DP solution procedure [1] implemented
by [10, 11], where the stratifyR package [13] has been customised further
to handle stratum costs. After substituting the value of ℎ−1 from equation
ℎ−1 = + + + ⋯ + ℎ−1 the following recurrence relations are
0
1
2
obtained. They are solved using the DP procedure which determines the
optimum solution by decomposing the MPP into stages, where each stage
comprises of a single variable subproblem. The recurrence relations of the DP
procedure in each case is given below:
For the first stage, = 1, at = :
∗
1 1
0 + 0
1
2
Φ = { ( + 1) [ (, ) − (, )]
1 1

+
× [ ( + 2, 0 ) − ( + 2, 1 0 )]

+ 2
2 2
− [ ( + 1, 0 ) − ( + 1, 1 0 )] × } (12)
ℎ

And for the stages ≥ 2:

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